Acta Mathematica Academiae Paedagogicae Ny´ıregyh´aziensis 33 (2017), 39–44
www.emis.de/journals ISSN 1786-0091
ON THE SOLVABILITY OF NON-HOMOGENEOUS STURM-LIOUVILLE PROBLEM
ANTON I. POPOV
Abstract. Non-homogeneous Sturm-Liouville problems can arise when trying to solve non-homogeneous partial differential equations or when con- structing the asymptotic series for partial differential equation solution.
The present paper gives a condition of solvability for the non-homogeneous Sturm-Liouville problem in general case for formal power series.
1. Introduction
Sturm-Liouville theory is a powerful instrument of the spectral theory. It is well described in many books (see, e.g. [5, 9] and references therein). Nu- merous physical problems (both quantum and classical) reduce to the Sturm- Liouville problem. One meet this problem when dealing with quantum wells, quantum graphs, wave guides, etc. [6, 7, 8, 11]. We mention also asymptoti- cal approach in waves theory. It is applied when one has a small parameter (coupling constant, perturbation parameter, etc.). Formally, an asymptotic approach reduces to construction of the asymptotic expansion in powers of this small parameter [1, 4, 10]. The series is constructed consequently, term by term. To find a term, it is necessary to solve the non-homogeneous Sturm- Liouville problem for formal power series with the right hand side depending on the previous terms. Correspondingly, the question appears about the solu- tion existence for this problem. One observe this situation, e.g. in asymptotic expansions related with space-time ray method [3, 12]. The present paper gives necessary and sufficient condition of solvability for the non-homogeneous Sturm-Liouville problem in general case.
2010Mathematics Subject Classification. 34B24, 34E05.
Key words and phrases. Sturm-Liouville problem, asymptotic expansion, power series.
This work was partially financially supported by the Government of the Russian Federa- tion (grant 074-U01), MK-5161.2016.1 of the President of the Russian Federation, by grant 16-11-10330 of Russian Science Foundation.
39
2. The main theorem
Theorem. Let us consider homogeneous Sturm-Liouville problem
(1)
Ly= (p(x)y′)′−q(x)y = 0, ℓ0y= (α0y+α1y′)
x=x0
= 0, ℓ1y= (β0y+β1y′)
x=x1
= 0.
(2) p(x)>0, Im q= 0.
αj, βj, j = 0,1 are real. p(x), q(x) are formal power series. Let there exist a solution y0 ̸= 0 in the form of a formal power series. Then the necessary and sufficient condition for the existence of the solution in the form of a formal power series of non-homogeneous Sturm-Liouville problem
(3)
Ly=−F, ℓ0y=A, ℓ1y=B.
is as follows:
(4) p(x1)B β1
y0 x=x1
−p(x0)A α1
y0 x=x0
=−
∫ x1
x0
F(x)y0(x)dx.
Here F(x), A, B is formal power series.
Proof. Necessary condition.
(5) (p(x)y′)′ −q(x)y=−F.
We multiply (5) by y0 and integrate from x0 tox1:
∫ x1
x0
((py′)′−qy)y0dx=
∫ x1
x0
(py′)′y0dx−
∫ x1
x0
qyy0dx (6)
=py′y0 x1
x0
−
∫ x1
x0
py′y0dx−
∫ x1
x0
qyy0dx
=py′y0 x1
x0
−py0′y x1
x0
+
∫ x1
x0
y(py0′)′dx−
∫ x1
x0
qyy0dx.
Then, we substitute the boundary conditions into (6):
(7) y0′(x0) = −α
α1y0(x0),
(8) y′(x0) = A
α1 − α
α1y(x0),
(9) y0′(x1) = −β0
β1y0(x1),
(10) y′(x1) = B
β1 − β0 β1y(x1).
Then∫ x1
x0
((py′)′−qy)y0dx=p(x1)B
β1y0(x1)−p(x1)β0
β1yy0(x1)
−p(x0)A α1
y0(x0) +p(x0)α0 α1
yy0(x0) +p(x1)β0 β1
y0y(x1) (11)
−p(x0)α0
α1y0y(x0) +
∫ x1
x0
y((py0′)′−qy0)dx
=p(x1)B β1y0
x=x1
−p(x0)A α1y0
x=x0
. From the other side,
(12)
∫ x1
x0
((py′)′−qy)y0dx=−
∫ x1
x0
F y0dx.
Equations (11) and (12) lead to (4), so we get necessary condition.
Sufficient condition.
Let us assume thatψ is a solution of the Cauchy problem:
(13)
(pψ′)′ −qψ=−F, ψ
x=x0
=A, ψ′
x=x0
=B.
The Cauchy problem always has a solution. Consequently, ψ exists. Let us considery=ψ−y0.
(14) (py′)′−qy = (pψ′)′−qψ−(py′0)′+qy0 =−F,
i.e. y satisfies the proper equation. Check the boundary conditions. We multiply the first equation in (13) by y0 and integrate from x0 tox1:
∫ x1
x0
((pψ′)′−qψ)y0dx=
∫ x1
x0
(pψ′)′y0dx−
∫ x1
x0
qψy0dx
=pψ′y0 x1
x0
−
∫ x1
x0
pψ′y0′dx−
∫ x1
x0
qψy0dx
=p(ψ′y0−y0′ψ) x1
x0
+
∫ x1
x0
ψ((py0′)′−qy0)dx
=p((y′+y′0)y0−y′0(y+y0)) x1
x0
=p(y′y0−y′0y) x1
x0
, (15)
so, (16)
∫ x1
x0
((pψ′)′−qψ)y0dx=p(x1)y′(x1)y0(x1)−p(x1)y0′(x1)y(x1)−
−p(x0)y′(x0)y0(x0) +p(x0)y0′(x0)y(x0).
We substitute (9)-(10) into (16) and come to the equation:
(17)
∫ x1
x0
((pψ′)′−qψ)y0dx=p(x1)y′(x1)y0(x1) +p(x1)β0
β1y0(x1)y(x1)−
−p(x0)y′(x0)y0(x0)−p(x0)α0
α1y0(x0)y(x0).
On the other side, (18)
∫ x1
x0
((py′)′−qy)y0dx=−
∫ x1
x0
F y0dx.
Let
(19) p(x1)B β1y0
x=x1
−p(x0)A α1y0
x=x0
=−
∫ x1
x0
F(x)y0(x)dx.
Then, relations (17)-(19) gives us:
(20) p(x1)B β1y0
x=x1
−p(x0)A α1y0
x=x0
=p(x1)y′(x1)y0(x1) +p(x1)β0
β1y0(x1)y(x1)−p(x0)y′(x0)y0(x0)−p(x0)α0
α1y0(x0)y(x0).
Condition (20) must be fulfilled for any x0 and x1. We fixx0, and will change x1. Since the ratio of (20) must always be performed, then parts of the equa- tion, corresponding to x0 and x1 should be independent of each other:
(21) −p(x0)A
α1y0(x0) = −p(x0)y′(x0)y0(x0)−p(x0)α0
α1y0(x0)y(x0).
(22) p(x1)B
β1y0(x1) =p(x1)y′(x1)y0(x1) +p(x1)β0
β1y0(x1)y(x1).
y0(x1) ̸= 0. Proof by contradiction. If y0(x1) = 0 then from the boundary condition ℓ1y= (β0y+β1y′)
x=x1
= 0 we get: y′0 x=x1
= 0. Consequently, y0 is
a solution of the Cauchy problem:
(p(x)y0′)′−q(x)y0 = 0, y0
x=x1
= 0, y′0
x=x1
= 0.
Therefore,y0 ≡0, which contradicts to the hypothesis of the theorem. Taking into account that p > 0, β1 ̸= 0, we can divide the both sides of (22) by p(x1)y0(x1) and multiply byβ1. As a result, we obtain:
(23) B =β1y′(x1) +β0y(x1), i.e. we come to the necessary condition for x=x1.
Let’s go back to relation (21):
(24) −p(x0)A
α1y0(x0) = −p(x0)y′(x0)y0(x0)−p(x0)α0
α1y0(x0)y(x0).
Taking into account that p > 0, y0(x0) ̸= 0 ((proved in a similar way as for y0(x1)), α1 ̸= 0, we come to the proper condition at x=x0:
(25) α1y′(x0) +α0y(x0) =A.
This completes the proof. □
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Received June 20, 2015.
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